Here, the function \(W = W(\mathbf {a}, \, \mathbf {d}, \, \xi )\)  is the so-called
renormalized energy, which accounts for the interaction between
the singular points \(a_j\) . It can be characterised in terms
of the Green function for the Laplace-Beltrami operator on \(M\) 
and the Gauss curvature of \(M\)  (see [1]}and section  below). When the \(u_\varepsilon \) 's
are minimizers of \(F_\varepsilon \) , \((\mathbf {a}, \, \mathbf {d}, \, \xi )\)  minimize the
renormalized energy \(W\) . The constant \(\gamma > 0\)  in (REF )
is the so-called core energy, which accounts for the energy
in a suitably small neighbourhood of each singular point \(a_j\) 
(we refer to [1]} for
the definition of \(\gamma \)  — in Ignat and Jerrard's notation,
\(\gamma = \iota _F\) ). Moreover, again up to a
to a subsequence, minimizers converge to a unit-norm vector field \(u_*\) ,
called a canonical harmonic vector field,
which is smooth in \(M\setminus \left\lbrace a_1, \, \ldots , \, a_n\right\rbrace \) 
and is uniquely determined by \((\mathbf {a}, \, \mathbf {d}, \, \xi )\)  up to a global rotation.
These results represent the extension of the classical two-dimensional results of Bethuel, Brezis and Hélein [3]}
to the case of vector fields on a Riemannian manifold (see also [4]} for a
\(\Gamma \) -convergence result).
